I got Tomb Raider in 1997 so not technically 'from the start' but pretty close, TR2 I only played the demo but that would have been around the release date.
Can't remember everything I found frustrating but the crux of it was
-Difficult to figure out what to do in some sections, in fact I think on TR2 I got stuck in the first cave of the demo for a while!
-Lot of random jumping about at ledges hoping I'd catch hold of one
-Controls felt awkward (possibly I could have fixed this with a different controller, who knows)
TR2013 on the other hand, you get instincts to help out when you can't figure out where to go, also Lara talks outloud in places, or there are other crutches to help you figure out where you need to get to (NPCs etc). Some might view this as dumbing-down or hand-holding, but personally running around for 10mins trying to figure out where I need to go does nothing for me, I wasted enough of my youth on such nonsense
The way it is implemented in TR2013 with instincts I really liked, it's not like some big HUD icon saying "OBJECTIVE 50m AWAY!!", so people can ignore it if they want.
Also in terms of graphics the original games were kinda limited in how diverse the environments could be, so you'd get a lot of things looking very 'samey' whereas in modern games the designer can highlight things a lot better in a non-contrived way, e.g. a ledge in Tomb Raider would basically be some blocky bit of rock sticking out, in TR2013 they can have pipework or something.
Then there is the fact that the original games were very sterile, if you weren't fighting you'd just be in a quiet room with just the orchestral music and the sound of Lara grunting and feet hitting the groud. TR2013 has environments that feel vibrant and alive, radio messages from NPCs, creatures roaming about etc. Playing the older games I felt so alone, the story seemed to play out through disjointed cutscenes rather than a fully integrated experience.